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2224-Sec12_4-HWT

# 2224-Sec12_4-HWT - Mat h 22 24 Multiva ri a ble C alc Sec...

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Math 2224 Multivariable Calc – Sec. 12.4: The Chain Rule I. Review from math 1205: The Chain Rule for functions of single variables A. The chain rule is used for composition functions: fog ( ) x ( ) = f g x ( ) ( ) B. Method 1: Rewrite the function in terms of y=f(u) and u=g(x). If y= f(u) and u=g(x), then dy dx = dy du ! " # \$ du dx ! " # \$ . C. Method 2: If y = fog ( ) x ( ) = f g x ( ) ( ) , then ! y = ! f g x ( ) ( ) ! g x ( ) D. Example: The radius of a right circular cylinder is increasing at a rate of 2 in min and the height is increasing at 3 in min . How fast is the lateral surface area changing when the radius is 10 inches and the height is 12 inches? II. The Chain Rule for Functions of Two or More Variables A. Functions of Two Variables 1. Theorem 5: The Chain Rule for Functions of Two Independent Variables If z = f ( x , y ) has continuous partial derivatives f x and f y and if x = x ( t ), y = y ( t ) are differentiable functions of t , then the composite z = f ( x ( t ), y ( t )) is a differentiable function of t and ! z ! t = f x ( x ( t ), y ( t )) " # x ( t ) + f y ( x ( t ), y ( t )) " # y ( t ) , or ! z ! t = ! z ! x " dx dt + ! z ! y " dy dt .

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